By letting \(a=\bar{f}(P,w,q)\)  and \(b=\bar{f}(S,u,q)\) , an \(\varepsilon \) -approximation implies that \(|a-b|\le \varepsilon \)  for every query \(q\) , and \(\varepsilon \) -net implies that \(b>0\)  if \(a\ge \varepsilon \) . Following [1]}, we define below a distance function that maps a positive real \(|a-b|_\nu \)  for each pair of positive real numbers \(a\)  and \(b\) . A specific value \(\nu \)  will imply \(|a-b|_{\nu }\le \varepsilon \) , i.e., that \(S\)  is an \(\varepsilon \) -approximation, and a different value of \(\nu \)  will imply that \(S\)  is an \(\varepsilon \) -net for \(P\) . This is formalized as follows.
